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Difference between revisions of "Zorn lemma"

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''maximal principle''
 
''maximal principle''
  
If in a non-empty [[Partially ordered set|partially ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993301.png" /> every totally ordered subset (cf. [[Totally ordered set|Totally ordered set]]) has an upper bound, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993302.png" /> contains a maximal element. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993303.png" /> is called an upper bound of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993304.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993305.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993306.png" />. If an upper bound for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993307.png" /> exists, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993308.png" /> is said to be bounded above. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z0993309.png" /> is called maximal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z09933010.png" /> if there is no element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z09933011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z09933012.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099330/z09933013.png" />.
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If in a non-empty [[Partially ordered set|partially ordered set]] $X$ every totally ordered subset (cf. [[Totally ordered set|Totally ordered set]]) has an upper bound, then $X$ contains a maximal element. An element $x_0$ is called an upper bound of a subset $A\subset X$ if $x\leq x_0$ for all $x\in A$. If an upper bound for $A$ exists, then the set $A$ is said to be bounded above. An element $x_0\in X$ is called maximal in $X$ if there is no element $x\in X$, $x\not=x_0$, such that $x_0\leq x$.
  
The lemma was stated and proved by M. Zorn in [[#References|[1]]]. It is equivalent to the [[Axiom of choice|axiom of choice]].
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The lemma was stated and proved by M. Zorn in {{Cite|Zo}}. It is equivalent to the [[Axiom of choice|axiom of choice]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Zorn,  "A remark on a method in transfinite algebra"  ''Bull. Amer. Math. Soc.'' , '''41'''  (1935)  pp. 667–670</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR></table>
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|valign="top"|{{Ref|Ke}}||valign="top"|  J.L. Kelley,  "General topology", Springer  (1975)  {{MR|0370454}}
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|valign="top"|{{Ref|Zo}}||valign="top"| M. Zorn,  "A remark on a method in transfinite algebra"  ''Bull. Amer. Math. Soc.'', '''41'''  (1935)  pp. 667–670 {{MR|1563165}}
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====Comments====
 
====Comments====
Earlier versions of the maximal principle, differing in detail from the one stated above but logically equivalent to it, were introduced independently by several mathematicians, the earliest being F. Hausdorff in 1909. For accounts of the history of the maximal principle, see [[#References|[a1]]][[#References|[a3]]].
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Earlier versions of the maximal principle, differing in detail from the one stated above but logically equivalent to it, were introduced independently by several mathematicians, the earliest being F. Hausdorff in 1909. For accounts of the history of the maximal principle, see {{Cite|Ca}}{{Cite|RuRu}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.J. Campbell,  "The origin of  "Zorn's lemma" "  ''Historia Math.'' , '''5'''  (1978)  pp. 77–89</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.H. Moore,  "Zermelo's axiom of choice" , Springer  (1982)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Rubin,  H. Rubin,  "Equivalents of the axiom of choice" , '''1–2''' , North-Holland  (1963–1985)</TD></TR></table>
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|valign="top"|{{Ref|Ca}}||valign="top"| P.J. Campbell,  "The origin of  "Zorn's lemma" "  ''Historia Math.'', '''5'''  (1978)  pp. 77–89 {{MR|0462876}}
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|valign="top"|{{Ref|Mo}}||valign="top"| G.H. Moore,  "Zermelo's axiom of choice", Springer  (1982) {{MR|0679315}}
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|valign="top"|{{Ref|RuRu}}||valign="top"| J. Rubin,  H. Rubin,  "Equivalents of the axiom of choice", '''1–2''', North-Holland  (1963–1985) {{MR|0153590}} {{MR|0798475}}
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Revision as of 21:28, 18 June 2014

2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]

maximal principle

If in a non-empty partially ordered set $X$ every totally ordered subset (cf. Totally ordered set) has an upper bound, then $X$ contains a maximal element. An element $x_0$ is called an upper bound of a subset $A\subset X$ if $x\leq x_0$ for all $x\in A$. If an upper bound for $A$ exists, then the set $A$ is said to be bounded above. An element $x_0\in X$ is called maximal in $X$ if there is no element $x\in X$, $x\not=x_0$, such that $x_0\leq x$.

The lemma was stated and proved by M. Zorn in [Zo]. It is equivalent to the axiom of choice.

References

[Ke] J.L. Kelley, "General topology", Springer (1975) MR0370454
[Zo] M. Zorn, "A remark on a method in transfinite algebra" Bull. Amer. Math. Soc., 41 (1935) pp. 667–670 MR1563165

Comments

Earlier versions of the maximal principle, differing in detail from the one stated above but logically equivalent to it, were introduced independently by several mathematicians, the earliest being F. Hausdorff in 1909. For accounts of the history of the maximal principle, see [Ca][RuRu].

References

[Ca] P.J. Campbell, "The origin of "Zorn's lemma" " Historia Math., 5 (1978) pp. 77–89 MR0462876
[Mo] G.H. Moore, "Zermelo's axiom of choice", Springer (1982) MR0679315
[RuRu] J. Rubin, H. Rubin, "Equivalents of the axiom of choice", 1–2, North-Holland (1963–1985) MR0153590 MR0798475
How to Cite This Entry:
Zorn lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zorn_lemma&oldid=11582
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article